Trigonometry (R-Formula)

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R-Formula: One Function Rules Them All

\begin{gathered}a\sin\theta \pm b\cos\theta = R\sin(\theta \pm \alpha)\\[2pt] a\cos\theta \pm b\sin\theta = R\cos(\theta \mp \alpha)\\[3pt] R = \sqrt{a^2+b^2}, \quad \tan\alpha = \tfrac{b}{a}\end{gathered}

One function = instant equations & max/min. Sine keeps the middle sign, cosine flips it.

Reflex: Let the leading function name the form; get α from tan⁻¹(second coeff/first coeff).

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Trigonometry (R-Formula)

3 worked examples

R-Formula: One Function Rules Them All

\begin{gathered}a\sin\theta \pm b\cos\theta = R\sin(\theta \pm \alpha)\\[2pt] a\cos\theta \pm b\sin\theta = R\cos(\theta \mp \alpha)\\[3pt] R = \sqrt{a^2+b^2}, \quad \tan\alpha = \tfrac{b}{a}\end{gathered}

One function = instant equations & max/min. Sine keeps the middle sign, cosine flips it.

Reflex: Let the leading function name the form; get α from tan⁻¹(second coeff/first coeff).

RHS Non-Zero? R-Formula, Not Tangent

5\sin(\theta + 53.130°) = 2

Dividing by $\cos\theta$ strands a $\frac{2}{\cos\theta}$ when RHS≠0 — rewrite the LHS as one function instead. Reflex: RHS zero → tangent trick. RHS non-zero → R-formula.

Max/Min Straight from the R-form

Max $=R+k$ at sine-angle $90°$ / cosine-angle $0°$ ; min $=-R+k$ at sine-angle $270°$ . Reflex: Check the angle is actually reachable in the given $\theta$ range — else the extreme shifts to an endpoint.

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